This discussion paper is/has been under review for the journal Hydrology and Earth SystemSciences (HESS). Please refer to the corresponding final paper in HESS if available.
Multiobjective sensitivity analysis andoptimization of a distributed hydrologicmodel MOBIDICJ. Yang1, F. Castelli2, and Y. Chen1
1State Key Laboratory of Desert and Oasis Ecology, Xinjiang Institute of Ecology andGeography, Chinese Academy of Sciences, Xinjiang, 830011, China2Department of Civil and Environmental Engineering, University of Florence, Italy
Received: 28 February 2014 – Accepted: 16 March 2014 – Published: 26 March 2014
Calibration of distributed hydrologic models usually involves how to deal with the largenumber of distributed parameters and optimization problems with multiple but oftenconflicting objectives which arise in a natural fashion. This study presents a multiob-jective sensitivity and optimization approach to handle these problems for a distributed5
hydrologic model MOBIDIC, which combines two sensitivity analysis techniques (Mor-ris method and State Dependent Parameter method) with a multiobjective optimization(MOO) approach ε-NSGAII. This approach was implemented to calibrate MOBIDICwith its application to the Davidson watershed, North Carolina with three objective func-tions, i.e., standardized root mean square error of logarithmic transformed discharge,10
water balance index, and mean absolute error of logarithmic transformed flow durationcurve, and its results were compared with those with a single objective optimization(SOO) with the traditional Nelder–Mead Simplex algorithm used in MOBIDIC by takingthe objective function as the Euclidean norm of these three objectives. Results show:(1) the two sensitivity analysis techniques are effective and efficient to determine the15
sensitive processes and insensitive parameters: surface runoff and evaporation arevery sensitive processes to all three objective functions, while groundwater recessionand soil hydraulic conductivity are not sensitive and were excluded in the optimization;(2) both MOO and SOO lead to acceptable simulations, e.g., for MOO, average Nash–Sutcliffe is 0.75 in the calibration period and 0.70 in the validation period; (3) evapora-20
tion and surface runoff shows similar importance to watershed water balance while thecontribution of baseflow can be ignored; (4) compared to SOO which was dependent ofinitial starting location, MOO provides more insight on parameter sensitivity and con-flicting characteristics of these objective functions. Multiobjective sensitivity analysisand optimization provides an alternative way for future MOBIDIC modelling.25
With the development of information technology (e.g., high performance computingcluster and remote sensing technology), there has been a prolific development of in-tegrated, distributed and physically-based watershed models (e.g., MIKE-SHE, Refs-gaard and Storm, 1995) over the past two decades, which are increasingly being used5
to support decisions about alternative management strategies in the areas of land usechange, climate change, water allocation, and pollution control. Though in principle pa-rameters of distributed and physically based models should be assessable from catch-ment data (in traditional conceptual rainfall–runoff models, parameters are obtainedthrough a calibration process), these models still need a parameter calibration process10
in practice due to scaling problems, experimental constraints, etc. (Beven and Binley,1992; Gupta et al., 1998; Madsen, 2003). Problems, arising in calibrating distributedhydrologic models, include how to handle large number of distributed parameters andoptimization problems with multiple but often conflicting objectives.
In the literature, to deal with large number of distributed model parameters, this is of-15
ten done by aggregating distributed parameters (e.g., Yang et al., 2007), or screeningout the unimportant parameters through a sensitivity analysis (e.g., Muleta and Nick-low, 2005; Yang, 2011). Sensitivity analysis can be used to not only screen out the mostinsensitive parameters, but also study the system behaviors identified by parametersand their interactions, qualitatively or quantitatively. However, most of applications in20
environmental modelling are based on the one-at-a-time (OAT) local sensitivity anal-ysis, which is “predicated on assumptions of model linearity which appear unjustifiedin the cases reviewed” (Saltelli and Annoni, 2010), or simple linear regressions wherea lot of uncertainty are not fairly accounted for. The use of global sensitivity analy-sis techniques is very crucial in distributed modelling. Only recently, global sensitivity25
analysis techniques started to appear in hydrologic modelling.Although most hydrologic applications are based on the single objective calibra-
tion, model calibration with multiple and often conflicting objectives arises in a natural
fashion in hydrologic modelling. This is not only due to the increasing availability ofmulti-variable (e.g., flow, groundwater level, etc.) or multi-site measurements, but alsodue to the intrinsic different system responses (e.g., peaks and baseflow in the flowseries). Instead of finding a single optimal solution in the single objective optimization(SOO), the task in the multiobjective optimization (MOO) is to identify a set of optimal5
trade-off solutions (called a Pareto set) between conflicting objectives. In hydrology, thetraditional method to solve multiobjective problems is to form a single objective, e.g.,by giving different weights to these multiple objectives or applying some transfer func-tion. Over the past decade, several MOO algorithms approaches have been appliedto the conceptual rainfall–runoff models (e.g., Yapo et al., 1998; Gupta et al., 1998;10
Madsen, 2000; Boyle et al., 2000; Vrugt et al., 2003; Liu and Sun, 2010), and nowincreasing applied to distributed hydrologic models (e.g., Madsen, 2003; Bekele andNicklow, 2007; Shafii and Smedt, 2009; MacLean et al., 2010). And there are somepapers (Tang et al., 2006; Wöhling et al., 2008) to comparatively study their strengthswith the application in hydrology. It is worth noting that the multiobjective calibration is15
different from statistical uncertainty analysis which is based on the concept (or similarconcept) of “equifinality” (see discussion in Gupta et al., 1998; Boyle et al., 2000).
This paper applies two sensitive analysis techniques (Morris method and State De-pendent Parameter method) and ε-NSGAII in the multiobjective sensitive analysis andcalibration framework. This was implemented to calibrate a distributed hydrological20
model MOBIDIC with its application to the Davidson watershed, North Carolina. Thepurpose is to study parameter sensitivity of the hydrologic model MOBIDIC and ex-plore the capability of MOO in calibrating the MOBIDIC compared to the traditionalSOO used in MOBIDIC applications.
This paper is structured as follows: Sect. 2 gives a description of the MOBIDIC25
model; Sect. 3 introduces the approach in the multiobjective sensitivity analysis andoptimization; Sect. 4 gives a brief introduction of the study site, model setup, objectiveselection, and sensitivity and calibration procedure; in Sect. 5, the results are presented
and discussed; and finally the main results are summarized and conclusions are drawnin “conclusions” section.
2 Hydrologic model MOBIDIC
MOBIDIC (MOdello di Bilancio Idrologico DIstribuito e Continuo; Castelli et al., 2009;Campo et al., 2006) is a distributed and raster-based hydrological balance model. MO-5
BIDIC simulates the energy and water balances on a cell basis within the watershed.Figure 1 gives a schematic representation of MOBIDIC. The energy balance is ap-proached by solving the heat diffusion equations in multiple layers in the soil–vegetationsystem, while the water balance is simulated in a series of reservoirs (i.e., boxes inFig. 1) and fluxes between them.10
For each cell, water in the soil is simulated by
dWg
dt= Inf −Sper −Qd −Sas
dWc
dt= Sas −Et (1)
where Wg [L] and Wc [L] are the water contents in the soil gravitational storage and15
capillary storage, respectively, and Inf [LT−1], Sper [LT−1], Qd [LT−1], Et [LT−1], and Sas
[LT−1] are infiltration, percolation, interflow, evaporation, and adsorption from gravita-tional to capillary storage, which are modeled through following equations:
where γ, β and κ are percolation coefficient [T−1], interflow coefficient [T−1], and soiladsorption coefficient [LT−1], respectively, P the precipitation [LT−1], Qh and Rd Hortonrunoff and Dunne runoff, Ks the soil hydraulic conductivity [LT−1], Wgmax [L] and Wcmax[L] the gravitational and capillary storage capacities.
Once the surface runoff (Qh and Rd) and baseflow are calculated, three different5
methods can be used for river routing, i.e., the lag method, the linear reservoir method,Muskingum–Cunge method (Cunge, 1969). Muskingum–Cunge method was used inthis study.
MOBIDIC uses either a linear reservoir or the Dupuit approximation to simulate thegroundwater balance which relates the groundwater change to the percolation, water10
loss in aquifers and baseflow. In this case study, the linear reservoir method was used.Although there are many distributed parameters in MOBIDIC, normally these dis-
tributed parameters are calibrated through the “aggregate” factors (e.g., the multiplierfor hydraulic conductivity) based on their initial estimations. And hereafter we use theterm “factor” (instead of “model parameter”) when we conduct the sensitivity analy-15
sis and optimization, to avoid the confusion with the term “model parameter” used inmodel description. A factor can be a model parameter or a group of model parameters,and in this paper it is a change to be applied to a group of model parameters. In MO-BIDIC, normally nine factors (i.e., nine groups of parameters) need to be calibrated.These factors, their explanations, and their corresponding model parameters are listed20
in Table 1.
3 Methodology
The procedure applied here consists of two-step analyses, i.e., a multiobjective sensi-tivity analysis generally characterizing the basic hydrologic processes and single outthe most insensitive parameters, and a multiobjective calibration aiming at trade-offs25
Sensitivity analysis is to assess how variations in model out can be apportioned, qual-itatively or quantitatively, to different sources of variations, and how the given modeldepends upon the information fed into it (Saltelli et al., 2008). In the literature, a lotof sensitivity analysis methods are introduced and applied, e.g., Yang (2011) applied5
and compared five different sensitivity analysis methods. Here we adopted an approachwhich combines two global sensitivity analysis techniques, i.e., the Morris method (Mor-ris, 1991) and SDP method (Ratto et al., 2007).
3.1.1 Morris method
Morris method is based on replicated and randomized one-factor-at-a-time design10
(Morris, 1991). For each factor Xi , Morris method uses two statistics, µi and σi , whichmeasure the degree of factor sensitivity, and the degree of nonlinearity or factor inter-action, respectively. The higher µi is, the more important the factor Xi is to the modeloutput; and the higher σi is, the more nonlinear the factor Xi is to the model output ormore interactions with other factors (details refer to Morris, 1991; Campolongo et al.,15
2007). Morris method takes m · (n+1) model runs to estimate these two sensitivity in-dices for each of n factors with sample size m. The advantage is it is efficient andeffective to screen out insensitive factors. Normally m takes values around 50. And ac-cording to Saltelli et al. (2008), the sensitivity measure (µi ) is a good proxy for the totaleffect (i.e., STi in Eq. (4) below), which is a robust measure in sensitivity analysis.20
3.1.2 State-Dependent Parameter method (SDP)
SDP (Ratto et al., 2007) is based on the ANOVA functional decomposition, which ap-portions the model output uncertainty (100 %, as 1 in Eq. 3) to factors and different
where Si is the main effect of factor Xi representing the average output variance reduc-tion that can be achieved when factor Xi is fixed, and Si j is the first-order interactionbetween Xi and Xj , and so on. In ANOVA based sensitivity analysis, total effect (STi )5
is frequently used, which stands for the average output variance that would remain aslong as Xi stays unknown,
STi = Si +∑j 6=i
Si j + . . .+S12...n (4)
SDP method uses the emulation technique to approximate lower order sensitivity in-dices in Eq. (3) (e.g., Si and Si j in this study) by ignoring the higher order sensitivity10
indices. And we define SDi = Si+∑jSi j (referred to as “quasi total effect” later) as a sur-
rogate to the total effect. The advantage is that it can precisely estimate lower ordersensitivity indices at a lower computational cost (normally 500 model runs, which isindependent of number of factors). The disadvantage is that it cannot estimate higherorder sensitivity indices.15
Since these two methods are computationally efficient, these two methods are ap-plied individually. And then, the sensitivity of each factor and its system behaviour willbe discussed, qualitatively by Morris method, and quantitatively by SDP method. Andthen the most insensitive factors will be screened out and excluded in the calibration.
In the context of multiobjective analysis, sensitivity analysis includes: (1) to examine20
the sensitivity of each factor to different objective functions, qualitatively or quantita-tively; (2) to single out the most sensitive factors and study the physical behavioursof the system; (3) to exclude the most insensitive factors and therefore simplify theprocess of calibration.
In the literature of hydrologic modelling, most applications are single objective based,which aims at a single optimal solution. However, for example in flow calibration, thereis always a case that two solutions, one solution better simulates the peaks and poorlysimulates the baseflow while the other solution poorly simulates the peaks while better5
simulates the baseflow. These two solutions, which are called Pareto solutions, are in-commensurable, i.e., better fitting of the peaks will lead to worse fitting of the baseflow,and vice versa. This belongs to the domain of MOO, aiming at finding a set of optimalsolutions (Pareto solutions), instead of one single solution.
Generally a MOO problem can be formulated as follows:10
min F (X) = (f1(X), f2(X), ..., fi (X), ..., fk(X))
Where X is an n-dimensional vector and in this study represents the model factors tobe calibrated, fi (X) i th objective function, and gi (X) i th constraint function.15
In the literature, there are many algorithms available to obtain the Pareto solutions,e.g., NSGAII (Non-dominated Sorting Genetic Algorithm-II; Deb et al., 2002), SPEA2(Strength Pareto Evolutionary Algorithm 2; Zitzler et al., 2001), MOSCEM-UA (Multi-objective Shuffled Complex Evolution Metropolis; Vrugt et al., 2003), and ε-NSGAII(Kollat and Reed, 2006), etc. In this study, we adopt ε-NSGAII, which is efficiency, re-20
liability, and ease-of-use. Its strengths have been comparatively studied in Kollot andRead (2006) and Tang et al. (2006).ε-NSGAII is an extension of the NSGAII (Deb et al., 2002), a second generation
of multiobjective evolution algorithm. The main characteristics of ε-NSGAII include: (i)selection, crossover, and mutation processes as other genetic algorithm by mimicking25
the process of natural evolution, (ii) an efficient non-domination sorting scheme, (iii)an elitist selection method that greatly aids in capturing Pareto front, (iv) ε-dominancearchiving, (v) adaptive population sizing, and (vi) automatic termination to minimize the
need for extensive parameter calibration. More details refer to Kollat and Reed (2006).As MOO is very time-consuming, we parallelized the source code and interfaced it withMOBIDIC.
As a comparison, a single objective function is defined as 2-norm of the multipleobjectives F (X), which measures how close to the original point (theoretical optimum5
O):
sof = ‖F (X)‖2 =
√√√√ k∑i=1
fi (X)2 (6)
And SOO was done with the classic Nelder–Mead algorithm (Nelder and Mead, 1965)which is widely used in MOBIDIC applications.
To analyze the Pareto solution and also compare with the solution from SOO, except10
for traditional methods, the “Level diagrams” proposed by Blasco et al. (2008) was alsoused. Compared to traditional methods, it can visualize high dimensional Pareto frontand synchronizes the objective and factor diagrams. The procedure and includes twosteps. In the first step, the vector of objectives (k-dimension) for each Pareto point ismapped to a real number (one-dimension) according to the proximity to the theoretical15
optimum measured with a specific norm of objectives; and in the second step, thesenorm values are plotted against the corresponding values of each objective or factor.1-norm, 2-norm and ∞-norm are suggested. To compare with SOO, 2-norm was used.
4 Davidson watershed and objective selection
4.1 Davidson watershed20
The Davidson watershed, located in southwest mountain area of North Carolina, drainsan area of 105 km2 above the station “Davidson river near Brevard” (see Fig. 2). The
elevation ranges from 645 m to 1820 ma.s.l. Based on the NLDAS climate data, the av-erage annual precipitation is 1900 mm and varies from 1400 mm to 2500 mm, and dailytemperature changes from −19 ◦C to 26 ◦C. The average daily flow is about 3.68 m3 s−1.
Data used in MOBIDIC model include (i) Digital Elevation Model (DEM), (ii) soil data,(iii) land cover data, (iv) climate data (precipitation, minimum and maximum tempera-5
ture, solar radiation, humidity and wind speed), and (v) flow data. 9 m DEM, land cover,SSURGO soil data, one station (Davidson river near Brevard) of flow data are fromUS Geological Survey, and hourly NLDAS climate data from National Aeronautics andSpace Administration (NASA). NLDAS integrates a large quantity of observation-basedand model reanalysis data to drive offline (not coupled to the atmosphere) land-surface10
models (LSMs), and executes at 1/8th-degree grid spacing over central North Amer-ica, enabled by the Land Information System (LIS) (Kumar et al., 2006; Peters-Lidardet al., 2007).
DEM is used to delineate the watershed and estimate the topographic parametersand river system, Land cover for evaporation parameters, soil data for soil parameters,15
climate data is used to drive MOBIDIC, and flow data are used to calibrate the modeland assess model performance. The climate and flow data used in this study are from1 January 1996 to 30 September 2006. As NLDAS only has hourly temperature dailyinstead of hourly minimum and maximum temperature needed by MOBIDIC, we com-piled the hourly climate data to daily data and run the model at a daily step. After20
MOBIDIC setup, the initial parameter values are listed in third column of Table 1.We split the data into a warm-up period (from 1 January 1996 to 30 September
2000), a calibration period (from 1 October 2000 to 30 September 2003), and a valida-tion period (from 1 October 2003 to 30 September 2006).
4.2 Objective function selection25
After setting up MOBIDIC in the Davidson watershed, three objective functions wereused in the multiobjective sensitivity analysis and optimization:
(1) Standardized root mean square error between the logarithms of simulated andobserved outflows:
SRMSE =
√1N
N∑i=1
(log
(Qobs
i
)− log
(Qsim
i
))2
√1
N−1
N∑i=1
(log
(Qobs
i
)− logQ
)2
(7)
(2) Water Balance Index, calculated as the mean absolute error between the simulatedand observed flow accumulation curves:5
WBI =1N
N∑i=1
∣∣∣QobsCi −Qsim
Ci
∣∣∣ (8)
(3) Mean absolute error between the logarithms of simulated and observed flow dura-tion curves
MARD =1
100
N∑i=1
∣∣∣log(Qobs
Pi
)− log
(Qsim
Pi
)∣∣∣ (9)
In Eqs. (7)–(9), Qobsi and Qsim
i are observed and simulated flow series at time step i , N10
the data length, logQ the average of logarithmic transformed observed flows, QobsCi and
QsimCi i th observed and simulated accumulated flows, and Qobs
Pi and QsimPi i th percentiles
of observed and simulated flow duration curves.SRMSE (Eq. 7), WBI (Eq. 8), and MARD (Eq. 9) are measures of the closeness
between simulated and observed flow series, water balance, and closeness between15
simulated and observed flow frequencies, respectively. The smaller these measuresare, the better the simulation is, and the minima are (0, 0, 0) meaning a perfect match
between the simulation and observation. It is worth noting that we use the logarithms ofthe flows instead of flows to avoid overfitting flow peaks (Boyle et al., 2000; Shafii andDe Smedt, 2009). And for SRMSE, we have NS ≈ 1−SRMSE2 when N is large (e.g.,> 100), where NS is the Nash–Sutcliffe coefficient (Nash and Sutcliffe, 1970), which iswidely used in hydrologic modelling.5
And accordingly, the single objective function here is the Euclidean norm (2-norm) ofSRMSE, WBI, and MARD:
sof =√
SRMSE2 +WBI2 +MARD2 (10)
5 Result and discussion
5.1 Multiobjective sensitivity analysis10
Morris method and SDP method were applied individually to the initially selected factors(in Table 1).
For Morris method, its convergences for three objective functions, monitored usingthe method proposed in Yang (2011), were achieved around 700 ∼ 800 model simu-lations. Figure 3 gives the sensitivity results for objective functions SRMSE, WBI, and15
MARD, respectively. In each plot, the horizontal axis (µ) denotes the degree of factorsensitivity, and the vertical axis (σ) denotes the degree of factor nonlinearity or interac-tion with other factors.
For SRMSE, the most sensitive parameters are group (pα, pγ, and pκ), followed bypβ and rCH, while other parameters (especially rKs and rKf) are not so sensitive. This20
applies to the degree of the factor nonlinearity or interaction. The sensitivities of pα, pγ,and pκ indicate the importance of their corresponding processes (i.e., surface runoff,percolation, and adsorption which is related to evapotranspiration) to SRMSE, whileinterflow (pβ) is less important and other processes/characteristics (e.g., groundwaterflow, rKf) are not important.25
For WBI, the dominating parameter is pκ, followed by group (pα, pγ, pβ and rCH),while other parameters (especially rKf and rWcmax) are not so sensitive. WBI measuresthe water balance between observed and simulated flow series, and it is reasonablethat pκ which controls the water supply for evaporation is most sensitive while otherfactors (pα, pγ, pβ and rCH) are sensitive mainly through interaction with this factor,5
as indicated by the high “σ”s of these factors.For MARD, the results are nearly the same to SRMSE.Figure 4 gives the sensitivity results based on SDP method for SRMSE, WBI, and
MARD, from top to bottom. In each plot, the grey and black bars are Si and SDi foreach factor.10
For SRMSE, as indicated by R2 in the legend, main effects (Si ) contribute to 58.7 %of SRMSE uncertainty, and quasi total effects (SDi ) account for 83 % of SRMSE uncer-tainty which is quite high, while other 17 % due to higher interactions are not explained.Based on SDi (black bar), the most sensitive factors are pγ and pκ, followed by pα andrCH, and then pβ and rWcmax while other factors are not sensitive. This result quantita-15
tively corroborates the result obtain from Morris method. The main effects (Si ) of (pγ,pκ, and pα) are high (i.e., 0.17, 0.18 and 0.14), which suggests these factors shouldbe determined first in model calibration as they lead to the largest reduction in SRMSEuncertainty. For each factor, the difference between the black bar and grey bar showsthe first order interaction with other factors. This interaction is very strong in pγ, pκ,20
pα, and rCH, and very weak in other factors.For WBI, as indicated by R2 in the legend, the total main effects (Si ) contribute to
38.4 % of the WBI uncertainty, quasi total effects (SDi ) only account for 57.6 % of WBIuncertainty, and 40 % due to higher interactions are not explained and can not beignored. However, by analyzing the result with that from Morris method (top right in25
Fig. 3), we still can get some valuable results: the dominating sensitive factor is pκ withSDi 0.43 (which is same as Morris method), followed by pγ, pα, and rCH, while otherfactors are not sensitive; the main effect of pκ is as high as 0.27, and it should be fixed
in order to get the maximum reduction in WBI uncertainty; the first interaction is high inpκ, pγ, and pα, not obvious in other factors.
Similar to the Morris results for SRMSE and MARD, the result of MARD is nearly thesame as SRMSE. The similar result for SRMSE and MARD shows similar characteristicrelationship between factors and the objective function. This is explainable: a good5
simulation measured by SRMSE will more likely result in a good measure of MARD,and vice versa.
As aforementioned, in the context of multiobjective sensitivity analysis, sensitivityanalysis is to exclude the factors which are insensitive to all the objective functionsconsidered. Based on the analysis above, four most insensitive factors are rKs, rKf,10
rWcmax,and rWgmax. However, as shown in Fig. 4, rWcmax is more sensitive than otherthree factors, and for objective function WBI, as higher order interactions are strongbased on SDP (i.e., explains around 40 % of model uncertainty), and evaporation isthe most sensitive process to water balance (as indicated by pκ and rCH) and rWcmaxis the only factor related to evaporation storage (Wc), therefore, we only exclude rKf,15
rKs, and rWgmax for calibration.
5.2 Multiobjective optimization
After sensitivity analysis, only six factors were involved in the calibration. MOO con-verged with 482 Pareto front points after totally 22 000 model runs with the parallelizedε-NSGAII, while SOO converged after 686 model runs with the classic Nelder–Mead20
algorithm. Apparently, ε-NSGAII took more model simulations than the Nelder–Meadalgorithm, but simulation time was compensated by the parallelized code running onhigh performance clusters.
Figure 5 shows optimized non-dominant sets normalized within [0, 1] and the blackline is for the factor set with SOO. It is encouraging that except rWcmax, factor ranges25
decreased a lot. This corroborates the conclusion in the sensitivity analysis: pγ, pκ,pβ, pα, and rCH are the most sensitive and identifiable factors to these three objec-tive functions, while rWcmax is less sensitive and less identifiable. Several scattered
values of pγ and dispersed rWgmax show that optimized factor sets are scattered in theresponse surface rather than concentrated in a continuous region. And the factor setwith SOO is within the range of non-dominant sets.
Figure 6 shows Pareto solutions scattered in the three-dimensional space (top left),and projections in two-dimensional subspaces with corresponding correlation coeffi-5
cients (r) in the calibration period, with the black dot in each plot denoting the solutionfor SOO. Correlation coefficients are high and negative for SRMSE and WBI (−0.54),and WBI and MARD (−0.74), and this indicates strong trade-off interactions along thePareto surface, i.e., better (lower) WBI will eventually result in worse (higher) SRMSE,and vice versa. The correlation coefficient is low (0.13) between SRMSE and MARD,10
and it is close to 0 when these two variables approach to their minima regions (i.e.,SRMSE < 0.53 and MARD < 0.09). Table 2 lists the statistics of these three objectivesassociated with Pareto sets and the result of SOO. For Pareto sets, in the calibrationperiod, the average SRMSE is 0.49 ranging from 0.47 to 0.57, which corresponds tothe average NS 0.78 ranging from 0.67 to 0.78; the average WBI is 0.05 ranging from15
0.02 to 0.11; and the average MEAD is 0.08 ranging from 0.03 to 0.11. In the validationperiod, the average SRMSE is 0.54 ranging from 0.51 to 0.62, which corresponds tothe average NS 0.70 ranging from 0.61 to 0.74; the average WBI is 0.05 ranging from0.04 to 0.09; and the average MEAD is 0.10 ranging from 0.08 to 0.13. And for SOO,SRMSE, WBI and MEAD are 0.48, 0.06 and 0.07 for the calibration period, and 0.57,20
0.06 and 0.10 for the validation, and accordingly the “NS”s are 0.77 and 0.67, respec-tively. According to Moriasi et al. (2007) which suggests NS> 0.75 and WBI< 10 % asexcellent modelling of river discharge, all Pareto solutions with MOO and the solutionwith SOO are close to “excellent” for both calibration and validation periods.
To better visualize Pareto sets and compare with the result of SOO, the level dia-25
grams are plotted in Fig. 7 by applying Euclidean norm (2-norm) to evaluate the dis-tance of each Pareto point to the ideal origin (0,0,0) (ideal values for all three normal-ized objectives are 0). In Fig. 7, top three plots are for three objectives and the rest foroptimized factors, and the black dot in each plot is the solution for SOO. In the level
diagrams, each objective and each factor of a point (corresponding to a Pareto solu-tion) is represented with the same 2-norm value for all the plots. Compared with MOO,obviously, SOO was trapped in the local optima as seen in top-left plot, which meansoptimization with Nelder–Mead algorithm was dependent of starting point. The 2-normhas a close linear relationship with SRMSE due to values of SRMSE are 5 to 10 times5
of other two objective functions, and it does not have such relationship with other twoobjectives. The scattering of objectives and factors makes it difficult in decision makingto select a single solution because there is not a clear trade-off solution (Blasco et al.,2008). However, compared to SOO, the Pareto solutions from MOO can make decisionmaking easy as it can be converted with expert opinion or some utility function.10
Figures 8 and 9 show simulated and observed flow duration curves and time seriesflows, respectively, with grey lines denoting the simulations with MOO and black lineswith SOO. Generally, all simulations match the observation well for both the durationcurve and time series flow for both calibration period and validation period. For the du-ration curve, simulations from MOO show a wide range in the low flows with frequencies15
from 0.85 to 1.0, which reflects the insensitivity of groundwater process (discussed inthe sensitivity analysis, i.e., rKf is insensitive to these three objectives). Except for this,there is a slight overestimation of flows, large flows during the calibration period withfrequencies from 0.2 to 0.1, and median to large flows during the validation period withfrequencies from 0.5 to 0.1. This might be due to the uncertainty in the reanalyzed cli-20
mate data. And the extreme flow with frequency around 0 is underestimated, and thisis because we chose the logarithm scale of the observed and simulated flows insteadof normal scale when computing objectives SRMSE and MARD. With SOO, the devia-tion from the observed is larger. Similar conclusions can be drawn from the time seriessimulations in Fig. 9, i.e., the wide ranges of low flow period, and underestimation of25
flow peaks. Other than this, generally all simulations can mimic the observations.Figure 10 shows the time series of watershed average storages (soil storage ex-
pressed as soil saturation, and groundwater depth), and fluxes (evaporation, surfacerunoff and baseflow) associated with MOO (shaded) and SOO (black line). With MOO,
soil saturation varies from 0.2 to 1.0 and groundwater from 0 to 120 mm. The tem-poral fluctuation of soil moisture is higher than groundwater, but lower than fluxes inevaporation and surface runoff. And this is true for the solution with SOO except theits ranges of soil saturation and groundwater (groundwater is very close to 0 mm). Forfluxes with MOO, evaporation and surface runoff have more temporal variation than5
baseflow, and their magnitudes are larger than baseflow. This applies to fluxes withSOO, and its baseflow is close to 0. This can be confirmed by the De Finetti diagram inFig. 11: with MOO, the average contributions of evaporation, surface runoff, and based-flow are 49.3, 46.1, and 4.8 %, respectively while the contribution of baseflow is veryinsignificant. And the contribution of baseflow is almost 0 with SOO.10
6 Conclusion
This study presents a multiobjective sensitivity and optimization approach to calibratea distributed hydrologic model MOBIDIC with its application in the Davidson watershedfor three objective functions (i.e., SRMSE, WBI, and MAED). Results show:
1. The two sensitivity analysis techniques are effective and efficient to determine the15
sensitive processes and insensitive parameters: surface runoff and evaporationare very sensitive processes to all three objective functions, while groundwaterrecession and soil hydraulic conductivity are not sensitive and were excluded inthe optimization.
2. For SRMSE and MAED, though all the factors have almost same sensitivities, the20
non-dominance of Pareto set shows the trade-off between these two objectives inthe response surface.
3. Both MOO and SOO achieved acceptable results for both calibration period andvalidation period, in terms of objective functions and visual match between simu-lated and observed flows and flow duration curves. For example, with MOO, the25
average NS is 0.75 ranging from 0.67 to 0.78 in the calibration and 0.70 rangingfrom 0.61 to 0.74 in the validation period.
4. In the case study, evaporation and surface runoff shows similar importance towatershed water balance while the contribution of baseflow can be ignored.
5. Compared to MOO with ε-NSGAII, the application of SOO with the Neld–Mead5
algorithm was dependent of initial starting point. Furthermore, the Pareto solu-tion provides a better understanding of these conflicting objectives and relationsbetween objectives and parameters, and a better way in decision making.
Acknowledgements. The research was supported by the “Thousand Youth Talents” Plan(Xinjiang Project) and the National Basic Research Program of China (973 Program:10
2010CB951003). The data used in this study were acquired as part of the mission of NASA’sEarth Science Division and archived and distributed by the Goddard Earth Sciences (GES)Data and Information Services Center (DISC).
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Table 1. Initial selected factors, initial estimation of the corresponding MOBIDIC parameter, andfactor ranges.
Factor Meaning of the given factor Initial estimation ofMOBIDIC parameter
Factor range
pγ Exponential changea on soil percolationcoefficient γ [s−1]
1.2×10−11 [−2, 9]
pκ Exponential change on soil adsorptioncoefficient κ [s−1]
1.6×10−7 [−6, 5]
pβ Exponential change on interflow coeffi-cient β [s−1]
2.5×10−6 [−7, 4]
pα Exponential change on surface storagedecay coefficient α [s−1]
3.3×10−7 [−6, 5]
rKs Multiplying changeb on soil hydraulicconductivity [ms−1]
[5.0×10−6,8.9×10−5] [0.001, 100]
rWcmax Multiplying change on maximum storageof the capillary reservoir [m]
[0.017, 0.165] [0.01, 5]
rWgmax Multiplying change on maximum storageof the gravitational reservoir [m]
[0.107, 0.449] [0.01, 5]
rCH Multiplying change on bulk turbulent ex-change coefficient for heat [–]
[0.010,0.018] [0.01, 5]
rKf Multiplying change on groundwater de-cay coefficient [s−1]
1.0×10−7 [0.001, 5]
a Exponential change pX means the corresponding MOBIDIC parameter X will be changed according toX = X0 ×exp(pX −1), where X0 is the initial estimation of X.b Multiplying change rX means the corresponding MOBIDIC parameter X will be changed according to X = X0 × rX.
discharge Qg, and surface runoff and interflow from upper cells (R+Qd)up), dashed arrows
different routings, and blue characters major model parameters.
Inf Ad
Pc
Qg
Qd
R
P
Wc (Wcmax)
Wg
(Wgmax)
Et (R+Qd)up
Groundwater storage H
River system
Ws α
CH
Ks β
γ
κ
Fig. 1. A schematic representation of MOBIDIC. Boxes denote different water storages (gravi-tational storage Wg, capillary storage Wc, groundwater storage H , surface storage Ws, and riversystem), solid arrows fluxes (evaporation Et, precipitation P , infiltration Inf, adsorption Ad, per-colation Pc, surface runoff R, interflow Qd, groundwater discharge Qg, and surface runoff andinterflow from upper cells (R +Qd)up), dashed arrows different routings, and blue charactersmajor model parameters.
Figure 6 The Pareto solutions in the three dimensional space (top left), and the
projections in the two dimensional subspace (other plots), with MOO, and the black dot is
the solution with SOO
0.480.5 0.520.540.560.05
0.1
0.04
0.06
0.08
0.1
SRMSEWBI
MAR
D
0.48 0.5 0.52 0.54 0.56
0.04
0.06
0.08
0.1
SRMSE
WBI
r =-0.54
0.48 0.5 0.52 0.54 0.56
0.04
0.06
0.08
0.1
SRMSE
MAR
D
r =0.13
0.04 0.06 0.08 0.1
0.04
0.06
0.08
0.1
WBI
MAR
D
r =-0.74
Fig. 6. The Pareto solutions in the three dimensional space (top left), and the projections in thetwo dimensional subspace (other plots), with MOO, and the black dot is the solution with SOO.
Figure 9 Observed flows (dotted) and simulated flows with MOO (grey) and SOO (black line) for the calibration period (top) and validation period (bottom)
Fig. 9. Observed flows (dotted) and simulated flows with MOO (grey) and SOO (black line) forthe calibration period (top) and validation period (bottom).
Figure 10 Time series of watershed average storages (soil water storage expressed as soil saturation, and groundwater depth), and fluxes (evaporation, surface runoff, and baseflow) with MOO (grey) and SOO (black line). For SOO, the groundwater storage and baseflow are close to 0 and hardly seen.
Fig. 10. Time series of watershed average storages (soil water storage expressed as soil sat-uration, and groundwater depth), and fluxes (evaporation, surface runoff, and baseflow) withMOO (grey) and SOO (black line). For SOO, the groundwater storage and baseflow are closeto 0 and hardly seen.
